Abstract
We consider a kind of nonsmooth optimization problems with l_{1}-norm minimization, which has many applications in compressed sensing, signal reconstruction, and the related engineering problems. Using smoothing approximate techniques, this kind of nonsmooth optimization problem can be transformed into a general unconstrained optimization problem, which can be solved by the proposed smoothing modified three-term conjugate gradient method. The smoothing modified three-term conjugate gradient method is based on Polak–Ribière–Polyak conjugate gradient method. For the Polak–Ribière–Polyak conjugate gradient method has good numerical properties, the proposed method possesses the sufficient descent property without any line searches, and it is also proved to be globally convergent. Finally, the numerical experiments show the efficiency of the proposed method.
Highlights
This problem is widely used in compressed sensing, signal reconstruction, analog-to-information conversion and related to many mathematical problems [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]
The transformation is based on the equivalence between a linear complementarity problem and an absolute value equation problem [17, 18]
Based on the above analysis, in this paper, we propose a new smoothing modified three-term conjugate gradient method to solve problem (1)
Summary
A smoothing gradient method has been given for solving problem (1) based on the new transformed absolute value equations in [14, 15]. The transformation is based on the equivalence between a linear complementarity problem and an absolute value equation problem [17, 18]. Based on the above analysis, in this paper, we propose a new smoothing modified three-term conjugate gradient method to solve problem (1).
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